Summary: THE ONE-DIMENSIONAL HUGHES MODEL FOR PEDESTRIAN
FLOW: RIEMANN­TYPE SOLUTIONS
DEBORA AMADORI AND MARCO DI FRANCESCO
Dedicated to Professor Constantine M. Dafermos for his 70th birthday
Abstract. This paper deals with a coupled system consisting of a scalar
conservation law and an eikonal equation, called the Hughes model. Introduced
in [24], this model attempts to describe the motion of pedestrians in a densely
crowded region, in which they are seen as a `thinking' (continuum) fluid. The
main mathematical difficulty is the discontinuous gradient of the solution to
the eikonal equation appearing in the flux of the conservation law. On a
one dimensional interval with zero Dirichlet conditions (the two edges of the
interval are interpreted as `targets'), the model can be decoupled in a way
to consider two classical conservation laws on two sub-domains separated by
a turning point at which the pedestrians change their direction. We shall
consider solutions with a possible jump discontinuity around the turning point.
For simplicity, we shall assume they are locally constant on both sides of the
discontinuity. We provide a detailed description of the local-in-time behavior
of the solution in terms of a `global' qualitative property of the pedestrian
density (that we call `relative evacuation rate'), which can be interpreted as
the attitude of the pedestrians to direct towards the left or the right target.